Hypergeometric functions over $\mathbb{F}_q$ and traces of Frobenius for elliptic curves

TL;DR

This paper establishes explicit links between Frobenius traces of elliptic curves and hypergeometric functions over finite fields, enhancing understanding of their arithmetic properties for specific prime power fields.

Contribution

It provides new explicit relations connecting Frobenius traces of elliptic curves with hypergeometric functions over finite fields for particular congruence classes of q.

Findings

Explicit formulas for Frobenius traces in terms of hypergeometric functions.

Connections established for q ≡ 1 mod 6 and q ≡ 1 mod 4.

Enhanced understanding of elliptic curve arithmetic over finite fields.

Abstract

We present here explicit relations between the traces of Frobenius endomorphisms of certain families of elliptic curves and special values of ${_{2}}F_1$-hypergeometric functions over $\mathbb{F}_q$ for $q \equiv 1 (\text{mod} 6)$ and $q \equiv 1 (\text{mod} 4)$.